If the equation $x^{4} + 4 x^{3} - 2 x^{2} - 12 x + a = 0$ has four distinct real roots, then the number of integral values of $a$ is

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Solution

$f (x) = x^{4} + 4 x^{3} - 2 x^{2} - 12 x + 9$
$\Rightarrow f (x) = (x^{2} + 2 x - 3)^{2}$

$\Rightarrow$ For four distinct roots,
$0 < 9 - a < 16$
$\Rightarrow - 16 < a - 9 < 0$
$\Rightarrow - 7 < a < 9$
$\Rightarrow$ Number of integral values of $a$ is $15$

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